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Unsteady rising of clean bubble in low viscosity liquid
José Manuel Gordillo, Benjamin Lalanne, Frederic Risso, Dominique
Legendre, Sébastien Tanguy
To cite this version:
José Manuel Gordillo, Benjamin Lalanne, Frederic Risso, Dominique Legendre, Sébastien Tanguy.
Unsteady rising of clean bubble in low viscosity liquid. Bubble Science, Engineering and Technology,
2012, vol. 4, pp. 4-11. �10.1179/1758897912Y.0000000002�. �hal-00920787�
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Eprints ID : 10560
To link to this article : DOI:10.1179/1758897912Y.0000000002
URL : http://dx.doi.org/10.1179/1758897912Y.0000000002
To cite this version : Gordillo, José Manuel and Lalanne, Benjamin
and Risso, Frederic and Legendre, Dominique and Tanguy, Sébastien
Unsteady rising of clean bubble in low viscosity liquid. (2012) Bubble
Science, Engineering and Technology, vol. 4 (n° 1). pp. 4-11. ISSN
1758-8960
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Unsteady rising of clean bubble in low
viscosity liquid
J. M. Gordillo*
1, B. Lalanne
2, F. Risso
2, D. Legendre
2and S. Tanguy
2When a submerged bubble is initially at rest in a stagnant low viscosity liquid such as water,
buoyancy forces accelerate the bubble upwards. The increasing relative velocity of the bubble
with the surrounding liquid provokes deformations on the bubble shape that affect its vertical
acceleration and also induce surface tension driven oscillations. Our theoretical model, which is
compared with full Navier–Stokes simulations predicts, with a reasonable accuracy, both the
position of the bubble centre of mass, as well as the time varying bubble shape under those
conditions for which the Reynolds number is large, the amplitude of the deformation is small, the
bubble interface is free of surfactants and the bubble rises following a straight vertical path. The
model can be used as a first approximation to describe the initial instants of the unsteady
buoyancy driven rising of millimetre sized bubbles typically generated in water aerators.
Keywords: Modelling, Numerical simulations
Introduction
Oxygenation of a liquid is a basic process of chemical industry which, in practice, is achieved by injecting either air or pure oxygen through nozzles or pipes placed at the bottom of a stagnant liquid pool. Clearly, the mass transfer process from the bubble towards the liquid occurs in the interval of time comprised between the instant of generation and the one at which the bubble meets the free interface. Since the bubble shape at detachment is rather different from that it possesses when it reaches the terminal velocity, the transition from the shape at generation and the steady, final shape, takes place through a series of oscillations. These oscillations are associated with two facts:
(i) the initial interfacial energy of the bubble is different from that it possesses when it reaches its terminal velocity
(ii) since the acceleration associated to buoyancy forces increases the bubble relative velocity with the surrounding medium, the time varying pressure distribution along the bubble interface induces changes in the bubble shape in order to meet the dynamic boundary condition at the bubble surface. It is well known that the virtual mass of an object depends on its shape, so the surface tension driven oscillations of the bubble provoke unsteady accelerations of its centre of mass.
The purpose of this paper is to derive from first principles a model describing the interfacial oscillations as well as the associated unsteady acceleration of the bubble centre of mass that takes place during the gravity driven rising of bubbles initially at rest. The model is constructed under the following hypotheses: the local Reynolds number is large, the relative deformation of the bubble is small, the bubble rises following a vertical a path and the bubble interface is clean, meaning that the shear stress at the bubble interface is zero. This latter assumption is the most restrictive one since, for it to be realistic, the liquid needs to be free of surfactants and this occurs only in those cases in which the liquid is either non-polar or water is ultrapurified.1
The same physical situation as the one considered here has constituted the subject of previous studies.2,3In those works, experiments were compared with the equations of motion obtained using force and energy balances. The system of equations derived in Refs. 2 and 3 requires a closing relationship related to the shape of the bubble, which is forced to be either the experimen-tally determined one or an ellipsoid. From a purely numerical point of view, Yang et al.4 reported full Navier–Stokes simulations to reproduce the rise and distortion of bubbles rising into a liquid bath whose pressure could be arbitrarily varied in time.
Although, as it has been described above, the acceleration from the rest of deformable bubbles has received some attention in the literature, none of the previous studies report equations describing, in a self-consistent manner, the coupling between the bubble acceleration and its deformation. To fill in this gap, it is reported here that a theoretical approximation, assum-ing small deformations of the bubble around its unperturbed spherical shape, provides two differential equations to describe the unsteady rising of bubbles in a low viscosity liquid free of surfactants.
1A´ rea de Meca´nica de Fluidos, Departamento de Ingenierı´a Aeroespacial
y Meca´nica de Fluidos, Universidad de Sevilla, Avda. de los Descubrimientos s/n, 41092, Sevilla, Spain
2Institut de Me´canique des Fluides de Toulouse, CNRS, Universite´ de
Toulouse (INP, UPS), Alle´e Camille Soula, 31400 Toulouse, France *Corresponding author, email jgordill@us.es
Theoretical description
The equations to be deduced next intend to model the unsteady vertical rising of millimetric air bubbles in pure water. Therefore, the typical values of the liquid density and liquid viscosity are r5103kg m23 and m51023Pa s21 respectively for a characteristic temperature of 20uC; the corresponding values for the gas are, at the same tem-perature, rg51?2 kg m23 and mg51?81025Pa s21. Since
the typical velocity of the bubble is given byyO(gR)1/2,1
with g and R denoting the gravitational acceleration and the unperturbed bubble radius respectively, the character-istic values of the Bond, Reynolds and Ohnesorge numbers for millimetric size bubbles [RyO(1) mm] rising in pure water are given by
Bo~r g R 2 s *0:1 Oh~ m rsR ð Þ1=2*10 {2 Re~r g Rð Þ 1=2 R m ~ Boð Þ 1=2Oh{1*30 (1)
with s50?072 N m21 the interfacial coefficient of pure water at 20uC. The model assumes that the interface is free of surfactants and the viscosity contrast between the liquid and the gas is
ml=mg*O(100)
consequently, the shear stress at the bubble interface is approximately zero.1Notice that the main consequence of considering a free slip boundary condition at the bubble interface is that, for the characteristic values of the dimensionless parameters given in equation (1), vorticity is confined to narrow regions, i.e. a boundary layer surrounding the bubble and a wake originated at its rear stagnation point extending downstream (see Fig. 1). Therefore, the flow is irrotational in the majority of the liquid bulk1and, thus, the velocity field in this large free
vorticity region, can be expressed as a function of a potential w. Moreover, since the viscous wake at the rear part of rising bubbles with sizes R 0:91 mm is stable,1the model assumes that the bubble rises following a vertical path. It will also be assumed that the bubble is axisym-metric and that it is only slightly deformed around its initial spherical shape. Finally, the consequence of the large density contrast existing between the liquid and the air is that inertial effects in the gas are neglected and, conse-quently, pressure is uniform inside the bubble.
The purpose of this paper is to derive a reduced order model to predict the rising velocity of the bubble centre of mass U(t), as well as the amplitude of the deformation s2(t) (with t denoting time), under the starting
hypoth-eses detailed above. In a frame of reference moving with the bubble dimensionless velocity U(t) (see Fig. 1), and using a system of spherical coordinates sketched in Fig. 1b, with the radial position r, the azimuthal angle h and the associated unit vectors erand eh, the amplitude
of the deformation s2(t) is defined from the equation that
approximately determines the position of the interface rs(h, t) as follows
rs~1zs2(t)P2(x) (2)
where x5cos h and Pi is the ith order Legendre
polynomial [P15x, P25(1/2)6(3x221)]. Notice that
equation (2) has been made dimensionless using the unperturbed bubble radius R and, also, that equa-tion (2) can be expressed in the following way
S(r,t,h)~rs{1{s2(t)P2(x)~0: (3)
The system of two equations needed to calculate the time evolution of the two dimensionless unknowns, namely, U(t) and s2(t), will be deduced applying global
balances of vertical momentum and kinetic energy to the region Vcdelimited by the surfaces S‘and Sbdepicted
in Fig. 1a. It is important to notice that the boundary at infinity S‘does not intersect the growing wake formed
at the rear part of the bubble and, thus, the Euler– Bernoulli equation can be used to relate the velocity potential with pressure in this region located very far
1 a sketch showing volume Vc delimited by surfaces S‘
and bubble interface Sb, where global balances of
energy and vertical momentum are applied in order to deduce simplified equations of motion: bubble rises at dimensionless velocity U(t) and g denotes gravitational acceleration; unit vector ez points upwards, in direction
opposite to gravity and coordinate z is vertical axis and b sketch showing system of spherical coordinates (r, h) with unit vectors er and eq in frame of reference
from the bubble interface. From now on all the variables are made dimensionless using the following scales: the vertical coordinate z (see Fig. 1) and the distance to the centre of the bubble r are made dimensionless using the unperturbed radius R, the velocity field u as well as the rising velocity U, are made dimensionless using (gR)1/2, the non-dimensional time t is made dimensionless using R/(gR)1/2, the non-dimensional velocity potential w is made dimensionless using R(gR)1/2and, finally, the scale used to obtain the non-dimensional values of the liquid pressure p, the gas pressure pg, the capillary pressure
Bo{1+:n (4)
where n is the outer normal to the bubble surface shown in Fig. 1, as well as the viscous stress at the interface
2Re{1n:c (5)
with
c~1=2 +uz(+u) T, (6) is rgR
The first step to derive the two desired equations is to express the velocity potential w, as a function of the two unknowns, namely, U(t) and s2(t). Note first that the
stress balance at the interface reads {pnz2Re{1n:cznp
g{Bo{1n+:n~0, (7)
i.e. tangential stress is zero at the interface of the bubble. This fact implies, as pointed out above, that vorticity is confined to narrow regions:1a boundary layer surround-ing the bubble and a wake originated at its rear stagnation point that extends downstream. This suggests to express the liquid velocity field as u5=wzDu, with Du?0 only in the aforementioned narrow rotational regions of the flow. Moreover, since the bubble volume is constant, the potential w can be expressed as an expansion in spherical harmonics of the form
w^a1(t) r2 P1(x)z
a2(t)
r3 P2(x) (8)
where only the first two terms in the spherical harmonics expansion of the potential have been retained. The relationship between the unknown functions a1and a2in
equation (8) with the two unknowns, is found by means of the kinematic boundary condition
LS
Ltz {U ef zz+w r~r½ s(h,t)g:+S~0 (9) with the function S defined in equation (3) and ez the
unitary vector pointing upwards depicted in Fig. 1. Indeed, using the equations for the potential w (equa-tion (8)) and S (equa(equa-tion (3)) one obtains
+w~{2a1 r3 P1er{ a1 r3sinheh{ 3a2 r4 P2er{ a2 r4sin hP’2eh LS Lt~{ : s2P2 +S~erzs2P’2sin heh (10)
with primes denoting derivatives with respect to x and dots denoting time derivatives. Once equation (10) is particularised at rs51zs2P2(x), substituted into
equa-tion (9) and the resulting expression is linearized using
s2,,1, the following equation relating ai(i51, 2) with s2
and U is obtained {s:2P2{U coshzUP02sin 2hs 2{2a1P1z 6a1P1P2s2{3a2P2z12a2P22s2{ {a1s2P02sin 2hz3a 2s2(P02) 2sin2h~0: (11)
By virtue of the orthogonality of Legendre polynomials with respect to the weight function sin h, i.e.
ðp 0
Pn(cosh)Pm(cosh)sin hdh~0 for n=m, (12)
the following equations for ai
a1~{ U 2 1{ 3 5s2 , a2~{ : s2 3 , (13)
have been deduced once equation (11) is multiplied by Pi
sin h and is integrated between 0 and p. Now, substituting equation (13) into equation (8), the expression obtained for the potential w as a function of both U and s2in the
laboratory frame of reference reads, w~{U 2r2 1{ 3s2 5 P1(x){ : s2 3r3P2(x): (14)
Once the potential w has been expressed as a function of the two unknowns, the equations for the time evolution of both the rising velocity U(t) and the deformation s2(t) are
obtained by means of the global balances of vertical momentum and kinetic energy applied to the control volume Vcin Fig. 1. It is crucial to note that the global
balances are referred to a laboratory frame of reference. Therefore, the two integral equations, which need to be solved subject to the dynamic boundary condition at the free interface (equation (7)) and to the kinematic condition
u:n~+w:n, Du:n~0, (15) read, respectively, ez: d dt ð Vc +wzDu ð Þdv~ ð Sb pn{2Re{1n:c dsz ð Vc {ezds{ ð S? npzn:uu ð Þds (16) and 1 2 d dt ð Vc +w:+wz2+w:DuzjDuj2 dv~ ð Sb u: pn{2Re {1n:cds{ ð Vc ez:uds{2Re{1 ð Vc c: c dv{ ð S? u:n pzjuj 2 2 ! ds: (17) Since {ez~{+z (18)
and due to the fact that the liquid is incompressible (=?u50, =?Du50), which implies that
+2w~0 (20)
+:(w+w)~+w:+w (21)
+:(Duw)~+w:Du (22) the volume integrals in the equations (16) and (17) can be transformed to surface integrals using equations (18)–(22) by means of the Gauss integral theorem.
Using the boundary conditions (7), (15) as well as equations (18)–(22), equation (16) can be expressed as
ez: d dt ð Sb {wnds~ ð Sb znds{d dt ð Vc Dudv (23) where it has been taken into account the additional fact that the net force on the bubble is zero, i.e
ð
Sb
{pgnzBo{1+:n
ds~0: (24)
The use of equations (7), (15) and (18)–(22) also leads to the following expression for the kinetic energy equa-tion (17) 1 2 d dt ð Sb {w+w:nds~{Bo{1 ð Sb +:n+w:ndsz ð Sb z+w:nds{ {2Re{1 ð Vc c: c dv{ ð Vc jDuj2 2 dv (25)
Let us point out that to deduce equations (23)–(25) it has been taken into account two additional facts: first, since wRr22and =wRr23for |r|R‘, the integral
ð
S?
w+w:n ds?0 (26)
and, second, due to the fact that the wake does not cross the outer boundary, Du50 at S‘and, since u5=wzDu
pzLw=Ltzjuj2=2zz~pzLw=Ltzj+wj2=2zz~C(t) (27) at S‘by virtue of the Euler–Bernoulli equation.5
Now, it only remains to calculate each of the integrals appearing in equations (23) and (25). For that purpose note first that, retaining only linear terms in s2, the
unitary normal vector to the interface Sb can be
approximated as
n^erzs2sinhP’2eh, (28)
from which it follows that
n:ez^cosh{s2P’2sin2h: (29)
Also, retaining only linear terms in s2, the differential
surface element ds in the surface integrals of equa-tions (23) and (25) can be approximated as follows
ds^2p sin h(1z2s2P2)dh: (30)
Using the expression of the potential w given by equation (14), together with the results given in the set of equations (28)–(30), the integrals at the left of equations (23) and (25) read respectively
ez: d dt ð Sb {wnds^2p 3 d dt U { 9 5Us2 (31) 1 2 d dt ð Sb {w+w:nds^4p 15 : s2 :: s2z 5 4 d dt U 21{9 5s2 , (32) where higher order terms in s2 have been neglected.
Moreover, notice that the interfacial curvature of the bubble surface
n~ +S
j+Sj, (33)
can be approximated, retaining only linear terms in s2, as
+n~2z4P2s2{10P22s22^2z4P2s2 (34)
Consequently, the power required to deform the bubble surface, represented by the first term at the right of equation (25), is given by {Bo{1 ð Sb +:n+w:nds~{16p 5 Bo {1s: 2s2: (35)
The calculation of the integrals associated to the effect of gravitational forces (first and second terms at the right of equations (23) and (25) respectively) leads to the following results ð Sb zez:nds~ 4p 3 , (36) ð Sb z+w:nds~4p 3 U , (37)
where it has been taken into account that z(rs,h)5(1zs2P2)cosh and have made use of the
equation for w (equation (14)). Finally, the power dissipated by viscous stresses (third integral at the right of equation (25)), namely
{2Re{1 ð
V
c: c dv, (38) can be easily calculated in the limit of high Reynolds numbers. Indeed, in those cases for which the Reynolds number is sufficiently large, the main contribution to the integral in equation (38) comes from the potential flow region.6 In such cases, this integral can be calculated following Batchelor5as
ðp 0
n:+(+w:+w)2p(1z2s2P2)sin hdh (39)
The integrand in equation (39) divided by 2p can be expressed as (1z2s2P2)n:+(+w:+w)jr~rS^{6 ( : s2)2P22z2U : s2P2cosh { {3 2 4 9( : s2)2(P02) 2 sin2hz4 3U : s2sin2hP02 {2s:2UP2cos h{ 2(s:2)2P22{ 1 3U : s2P02sin 2h{2 9( : s2)2(P02)
2sin2h{6U2cos2h{
3 2U 2sin2 hz42U2s2P2cos2hz 36 5 U 2s 2cos2hz 21 2 U 2s 2P2sin2hz 18 10U 2s 2sin2h{ 3 2U 2s 2P02sin 2hcosh{
12U2s2P2cos2h{3U2s2P2sin2h: (40)
and, therefore, the power dissipated by viscous stresses is approximately given, in the limit Re&1, substituting equation (40) into equation (38)
{2Re{1 ð V c: c dv~ {12Re{1pU2ð1{2s2Þ{ 32p 3 Re {1s:2 2: (41)
Consequently, the substitution of equations (31), (32), (35)–(37) and (41) into equations (23)–(25), leads to the following expressions for the vertical momentum and kinetic energy equations
2p 3 d dt U { 9 5s2U ~4p 3 zFv (42) with Fv~{ d dt ð Vc ez:Dudv (43) and 4p 15 d dt : s2 2 2 z 5 4 U 2 1{9 5s2 z6Bo{1s22 ~ 4p 3 U { 1 Re 12pU 2 1{2s 2 ð Þz32p 3 : s22 : (44) Substracting from equation (44) the result of multi-plying equation (42) by U, the following couple of equations for both U and s2is obtained
d dt U { 9 5s2U ~2{18U Re{1(1{2s2), (45) and d2s2 dt2 z40Re {1ds2 dtz12Bo {1s 2~{ 9 4U 2 : (46) To deduce equations (45) and (46), Fvis not calculated,
but identified with those terms proportional to URe21. The kinetic energy contained in the regions where vorticity is different from zero (fourth integral at the right of equation (25)) has also been neglected. It will be discussed below that the system (45) and (46) is quite similar, albeit not identical, to that obtained by Doinikov,12 who deduced his result by means of a different mathematical technique. Also notice that the terms representing both the natural oscillation frequency (12Bo21)1/2 in equation (46) as well as the damping coefficient (40Re21) in the same equation, coincide with
the values for a bubble at rest13in spite of the bubble’s centre of mass velocity is different from zero.14
Results
The results of the model are compared with direct numerical simulations performed with two different codes solving the full Navier–Stokes equations but based on two different methods for the interface capture. In DIVA,7,8a level set ghost fluid method is used while in JADIM9–11a volume of fluid method with no interface reconstruction is implemented. The first check of the theory is depicted in Fig. 2, where the theoretical dimensionless added mass coefficient namely CM51/
2(129/5s2) is compared with the one calculated with each
of the numerical codes. The theoretical result, which has been deduced neglecting higher order terms in s2, agrees
well with numerical results for the smallest values of s2
but, as expected, the agreement deteriorates for s2 0:2.
In Figs. 3–5, the time evolutions of both U(t) and s2(t)
predicted by equations (45) and (46) are compared with the ones computed numerically for several values of Bo and a fixed value of the Ohnesorge number
Oh~ Boð Þ1=2=Re~0:01:
In the case of the smallest Bo considered, Bo50?0164, Fig. 3 shows that the time evolution of the bubble deformation is very well captured by the model. However, the ascending velocity is only reproduced during the initial instants of the acceleration process. There are two reasons that explain the mismatch between the time evolution of the centre of mass velocity U(t), predicted by the set of equations (45) and (46) and the numerical result. One of them is that the theory developed here only retains linear terms in s2. However, the main reason for
the disagreement between theory and numerics depicted in Figs. 3–5 is associated to the fact that the theoretical results deduced here are only valid in the limit Re&1. Indeed, the fourth integral on the right hand side of equation (25) has been neglected, so the viscous dissipa-tion that takes place into both the unsteady boundary layer and the unsteady wake is not taken into account by the model. The calculation of the dissipation in these regions where vorticity is confined, would lead to a generalised memory force term in equations (45) and (46) that would clearly improve the results, but the correct quantification of this new term is out of the scope of this study. Nevertheless, it can be checked whether the addi-tion to the term proporaddi-tional to Re21in equation (45) of theyO(Re23/2) viscous correction term due to Moore,6
improves the comparison between theory and numerical results or not. Notice, however, that the origin of Moore’s correction6corresponds to the viscous dissipation taking place into both the boundary layer and the wake of a bubble moving steadily. In spite that the type of flow in the physical situation studied here is unsteady, if the term proportional to Re21 in equation (45) is substituted by Moore’s result, i.e.
2 Comparison between dimensionless added mass coeffi-cient deduced in model 1/2(129/5s2) and the ones
com-puted numerically using DIVA7,8 or JADIM9–11 codes for
several values of Bond and Ohnesorge numbers
defined in equation (1)
3 Theoretical time evolution of vertical velocity U(t) and bubble distortion s2(t) compared with numerical results for
Bo50?0164 and Oh5Bo1/2/Re50?01: results of model have been calculated including and not including Moore’s
correc-tion term, whereas numerical simulacorrec-tion has been performed using DIVA;7,8 in this case, addition of viscous term by Moore,6 improves prediction of terminal velocity; results corresponding to JADIM9–11 code have not been included
since magnitude of numerical spurious currents is important when compared to velocity of bubble
4 Theoretical time evolution of vertical velocity U(t) and bubble distortion s2(t) compared with numerical results for Bo50?0945
and Oh5Bo1/2/Re50?01: results of model have been calculated including and not including Moore’s correction term, whereas numerical simulation has been performed using either DIVA7,8or JADIM9–11numerical codes; agreement between
theory and numerics is not only fairly good during initial stages of acceleration process, but also at intermediate times
5 Theoretical time evolution of vertical velocity U(t) and bubble distortion s2(t) compared with numerical results for
Bo50?2625 and Oh5Bo1/2/Re50?01: results of model have been calculated including and not including Moore’s correction
term, whereas numerical simulation has been performed using either DIVA7,8or JADIM9–11numerical codes; For this large
18 U Re{1(1{2s2)| 1{2:2= 2ReUð Þ1=2
h i
the agreement with numerical results notably improves in the case depicted in Fig. 3. Indeed, notice that the terminal velocity predicted by the model is closer to the numerical result than that in the case of theyO(Re23/2) term [22?2/(2ReU)1/2] is neglected.
The fact that the equations (45) and (46) have been deduced retaining only linear terms in s2, also poses
limitations to the theory. This is clearly seen in Fig. 5, where the results corresponding to the larger value of Bo are depicted: since the bubble deformation s2 increases
with Bo, the agreement depicted in Fig. 5 between theory and the numerics, is the poorest of the three cases considered. Notice also that the agreement between the numerics and theory in the case depicted in Fig. 4, which corresponds to the case of bubbles with R^1 mm, is reasonable for all times. This is due to the fact that the starting hypotheses with which the present model has been deduced correspond, precisely, to this bubble size. Let us conclude stating that the agreement between theory and numerics is fairly good during the initial stages of the acceleration process with independence of the value of Bo considered. However, the good agreement deteriorates for larger times due to the fact that the model does not appropriately account for the viscous dissipation in the narrow time varying regions where vorticity is confined, namely, the boundary layer and the wake.
Conclusions
This paper presents a model that can be used as a first approximation to describe the initial instants of the unsteady buoyancy driven rising of millimetre sized bubbles generated in water aerators. More precisely, the authors have deduced an approximate theoretical model which includes two equations to describe the time evolutions of both the vertical velocity and the distor-tions experienced by a clean bubble which accelerates from the rest due to the action of buoyancy forces. The theoretical analysis developed here assumes that the bubble deformation is small and that the Reynolds number is sufficiently large so as to neglect the dissipation in both the unsteady boundary layer and the wake. The main contribution of this paper is that the reduced order model deduced retains, in a self-consistent manner, the effects on the virtual mass of the bubble of the distortions created by the increasing relative velocity of the bubble with the surrounding liquid and by capillary oscillations. In addition, the terms associated with viscosity are correctly calculated up to first order in the limit of high Reynolds numbers and small deforma-tions. The main differences of the theory developed here with the ones presented previously are the following. Differently to van Wijngaarden and Veldhuis2 it is not assumed here that the bubble is ellipsoidal, viscosity is retained in the present analysis up to first order and the effect of the bubble centre of mass velocity on the bubble oscillations is also taken into account. In the paper by de Vries and Lohse,3 the bubble is also assumed to be ellipsoidal but, differently to the theoretical approach by van Wijngaarden and Veldhuis,2both the viscous drag and the effect of the bubble velocity on its shape, are retained in their model. However, the theory by de Vries
and Lohse3 needs to be fed with the experimentally measured aspect ratio of the bubbles. In Yang et al.,4the authors simplified the analysis considering that the rising bubble is spherical. Their model retained the effect of the history force through a term which is only valid for the case of spherical bubbles moving at low Reynolds numbers. In spite of the addition of such term is not justified in the range of Reynolds numbers considered in their study, Yang et al.4 found that its inclusion improves the comparison with direct numerical simula-tions. Finally, Doinikov12deduced, in a self-consistent manner but neglecting viscosity, a system of equations for both the vertical position of the bubble centre of mass and the shape oscillations retaining four terms in the expansion of the potential w in spherical harmonics. Following a different mathematical technique, the author found a system of ordinary differential equations very similar to that deduced here. However, in Doinikov’s paper,12 viscous terms are included ad hoc
and only in the equation that represents the vertical rising velocity (equation (45) for U(t) here). In addition, the viscous term added ad hoc by Doinikov assumes that the bubble is spherical. This is not consistent with the potential flow part of the analysis, where it is considered that the bubble is deformable.
The comparison with full Navier Stokes simulations using two different numerical codes reveal that with independence of the Bo number, the agreement between theory and numerics is fairly good during the initial instants of time. However, the good agreement deterio-rates for larger values of the dimensionless time. This indicates that the inertia of the liquid contained in the viscous regions, which would appear in equations (45) and (46) in the form of a generalised history force of the type included in their model by Yang et al.4cannot be neglected in the analysis. An alternative to account for the viscous dissipation in the regions where vorticity is confined, would be to follow the procedure in Legendre et al.,15 where the history force term is calculated numerically. The other possibility would be to follow an analytical approach. This latter option implies to describe the time evolution of the vorticity in the narrow regions where it is confined. This would lead to a set of boundary layer equations for the flow around the bubble and the wake, a non-trivial calculation which is out of the scope of this preliminary study.
Acknowledgements
J. M. Gordillo thanks financial support by the Spanish Ministry of Education under project DPI2011-28356-C03-01 and the Junta de Andalucı´a under project P08-TEP-03997.
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